Circular, Stationary Profiles Emerging in Unidirectional Abrasion

Abstract

Models of moving interfaces, especially those with stable soliton-like behavior, are of central importance in many areas of physics, for example, surface growth models, chemical waves, and more. Here described for the first time in a realistic geophysical context is a partial differential equation model of bedrock abrasion by unidirectional impacts generalizing earlier pioneering work on pebble shapes by Bloore who treated isotropic impacts (Bloore in Math. Geol. 9:113–122, 1977). The result is a simple geometrical partial differential equation exhibiting circular arcs as solitary wave profiles. The latter seem to be the first known analytic solutions on Bloore-type models. Solitonic behavior, although familiar in many areas of physics, appears not to have been encountered in the geophysical literature. Not only are the existence and stability of these stationary, traveling shapes demonstrated here by numerical experiments based on finite difference approximations, but it is also shown that the results received here are consistent with recent laboratory experiments. The simulations within show that, depending on initial profile shape and other parameters, these circular profiles may evolve via long transients, which in a geological setting, may appear as noncircular stationary profiles.